Dynamic Data Structures for Interval Coloring

Abstract

We consider the dynamic graph coloring problem restricted to the class of interval graphs. At each update step the algorithm is presented with an interval to be colored, or a previously colored interval to delete. The goal of the algorithm is to efficiently maintain a proper coloring of the intervals with as few colors as possible by an online algorithm. In the incremental model, each update step presents the algorithm with an interval to be colored. The problem is closely connected to the online vertex coloring problem of interval graphs for which the Kierstead-Trotter (KT) algorithm achieves the best possible competitive ratio. We first show that a sub-quadratic time direct implementation of the KT-algorithm is unlikely to exist conditioned on the correctness of the Online Boolean Matrix Vector multiplication conjecture due to Henzinger et al. \cite{DBLP:conf/stoc/HenzingerKNS15}. We then design an incremental algorithm that is subtly different from the KT-algorithm and uses at most $3 \omega - 2$ colors, where $\omega$ is the maximum clique in the interval graph associated with the set of intervals. Our incremental data structure maintains a proper coloring in amortized $O(\log n + \Delta)$ update time where $n$ is the total number of intervals inserted and $\Delta$ is the maximum degree of a vertex in the interval graph. We then consider the fully dynamic framework involving insertions and deletions. On each update, our aim is to maintain a $3 \omega - 2$ coloring of the remaining set of intervals, where $\omega$ is the maximum clique in the interval graph associated with the remaining set of intervals. Our fully dynamic algorithm supports insertion of an interval in $O(\log n + \Delta \log \omega)$ worst case update time and deletion of an interval in $O(\Delta^2 \log n)$ worst case update time.